Wavefield extrapolation and simulation (and wave equation solutions in general) are used in seismic modeling, imaging, and inversion and they tend to consume 90% of the computational effort in the Oil and Gas exploration industry. The process is governed by the wave equation and uses knowledge of the medium elastic and anisotropic (including in some cases poroelastic) properties. Since seismic data are recorded using sensors distributed over a limited region (for example, the Earth's surface) of the domain of interest, wavefield extrapolation by solving the wave equation is needed, along with the medium properties. Recently, with high-resolution data, the industry has recognized that anisotropic behavior of wavefields is prevalent in seismic data. Thus, accurate simulation of wave propagation in the subsurface includes anisotropy in wave extrapolation. However, the cost of wavefield extrapolation in inhomogeneous anisotropic media exceeds that in isotropic inhomogeneous media. In fact, for a transversely isotropic media with a tilted axis of symmetry the cost could be five fold. Add to that, in the Gulf of Mexico about 70% of the reverse time migration applied there is, now, based on a tilted transversely isotropic media (TTI) description of the medium.